Suppose that the mean cranial capacity for men is 1090 cc (cubic centimeters) and that the standard deviation is 300 cc. Assuming that men's cranial capacities are normally distributed, complete the following statements.(a) Approximately 95% of men have cranial capacities between cc and cc.(b) Approximately of men have cranial capacities between 790 cc and 1390 cc.
Question
Suppose that the mean cranial capacity for men is 1090 cc (cubic centimeters) and that the standard deviation is 300 cc. Assuming that men's cranial capacities are normally distributed, complete the following statements.(a) Approximately 95% of men have cranial capacities between cc and cc.(b) Approximately of men have cranial capacities between 790 cc and 1390 cc.
Solution
(a) The 95% of any normally distributed data set lies within 2 standard deviations of the mean. So, to find the cranial capacities that 95% of men have, we need to add and subtract 2 standard deviations from the mean.
Mean = 1090 cc Standard Deviation = 300 cc
Lower limit = Mean - 2Standard Deviation = 1090 cc - 2300 cc = 490 cc Upper limit = Mean + 2Standard Deviation = 1090 cc + 2300 cc = 1690 cc
So, approximately 95% of men have cranial capacities between 490 cc and 1690 cc.
(b) To find the percentage of men who have cranial capacities between 790 cc and 1390 cc, we need to calculate the z-scores for these values and look up the corresponding percentages in a standard normal distribution table.
Z-score for 790 cc = (790 cc - Mean) / Standard Deviation = (790 cc - 1090 cc) / 300 cc = -1 Z-score for 1390 cc = (1390 cc - Mean) / Standard Deviation = (1390 cc - 1090 cc) / 300 cc = 1
Looking up these z-scores in a standard normal distribution table, we find that approximately 34.13% of data lies between the mean and 1 standard deviation on either side of the mean. So, the percentage of men who have cranial capacities between 790 cc and 1390 cc is approximately 34.13% + 34.13% = 68.26%.
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